for \(i=1,\ldots ,N\) ,
where \(F_i\colon \mathcal {X}\rightarrow \mathcal {X}\)  and \(G_{i,k}\colon \mathcal {X}\times \mathcal {X}\rightarrow \mathcal {X}\)  represent the individual dynamics of the element \(i\)  and effects of the element \(k\)  on the element \(i\) , respectively.
It is often the case that the intrinsic frequencies of the elements are close to each other [1]}, [2]}. Thus, we here assume that the elements \(1,\ldots ,N\)  are dominated by a common intrinsic frequency \(\omega \)  and weakly interact. That is, we consider the situation where the dynamics can be reduced to the following phase model:
\(\begin{aligned}\frac{d\theta _{i}(t)}{dt}=\omega +&\Gamma _{i}(\theta _{i}(t)-\theta _{1}(t),\ldots ,\theta _{i}(t)-\theta _{N}(t)) \end{aligned}\) 
